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%I #3 Mar 30 2012 18:37:03
%S 1,1,1,4,3,1,17,8,5,1,98,41,12,7,1,622,234,73,16,9,1,4512,1602,418,
%T 113,20,11,1,35373,11976,3110,650,161,24,13,1,300974,98541,23920,5242,
%U 930,217,28,15,1,2722070,866942,207549,41304,8094,1258,281,32,17,1
%N Triangle, read by rows, where T(n,k) equals the dot product of the vector of terms in row n that are to the right of T(n,k) with the vector of terms in column k that are above T(n,k) for n>k+1>0, with the odd numbers in the secondary diagonal and all 1's in the main diagonal.
%F T(n,k) = Sum_{j=0..n-1-k} T(n,k+j+1)*T(k+j,k) for n>k+1>0, with T(k,k) = 1 and T(k+1,k) = 2k+1 for k>=0.
%e Illustrate the recurrence by:
%e T(n,k)=[T(n,k+1),T(n,k+2),..,T(n,n)]*[T(k,k),T(k+1,k),..,T(n-1,k)]:
%e T(3,0) = [8,5,1]*[1,1,4]~ = 8*1 + 5*1 + 1*4 = 17;
%e T(4,1) = [12,7,1]*[1,3,8]~ = 12*1 + 7*3 + 1*8 = 41;
%e T(5,1) = [73,16,9,1]*[1,3,8,41]~ = 73*1 + 16*3 + 9*8 + 1*41 = 234;
%e T(6,3) = [113,20,11,1]*[1,5,12,73]~ = 113*1 + 20*5 + 11*12 + 1*73 = 418.
%e Triangle begins:
%e 1;
%e 1, 1;
%e 4, 3, 1;
%e 17, 8, 5, 1;
%e 98, 41, 12, 7, 1;
%e 622, 234, 73, 16, 9, 1;
%e 4512, 1602, 418, 113, 20, 11, 1;
%e 35373, 11976, 3110, 650, 161, 24, 13, 1;
%e 300974, 98541, 23920, 5242, 930, 217, 28, 15, 1;
%e 2722070, 866942, 207549, 41304, 8094, 1258, 281, 32, 17, 1;
%e 26118056, 8139602, 1885166, 377757, 65088, 11762, 1634, 353, 36, 19, 1;
%o (PARI) {T(n,k)=if(n==k,1,if(n==k+1,2*n-1,sum(i=0,n-k-1,T(n,k+i+1)*T(k+i,k))))}
%Y Cf. A128321 (column 0), A128322 (column 1), A128323 (column 2), A128324 (row sums); variant: A115080.
%K nonn,tabl
%O 0,4
%A _Paul D. Hanna_, Feb 25 2007
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